The first thing that sprang to mind for me is the possibility of (corporate) policy restricting or denying self signed certificates. Obviously though that would only apply to https connections, and is really just a guess.

If you give a bit more specific info by answering Don's questions hopefully we can help you sort this out.

So redmine seem to be fine, but projectpier requires the check on remember me. I'm going to go back to the users who had difficulty with redmine and IE to see if their login problem with redmine is valid.